Geometry of the Projectivization of Ideals and Applications to Problems of Birationality Remi Bignalet-Cazalet
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ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 (1) Let X Be a Scheme and Z a Closed Subset of X. Show That There Is a Closed Imme
ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 CAUCHER BIRKAR (1) Let X be a scheme and Z a closed subset of X. Show that there is a closed immersion f : Y ! X which is a homeomorphism of Y onto Z. (2) Show that a scheme X is affine iff there are b1; : : : ; bn 2 OX (X) such that each D(bi) is affine and the ideal generated by all the bi is OX (X) (see [Hartshorne, II, exercise 2.17]). (3) Let X be a topological space and let OX to be the constant sheaf associated to Z. Show that (X; OX ) is a ringed space and that every sheaf on X is in a natural way an OX -module. (4) Let (X; OX ) be a ringed space. Show that the kernel and image of a morphism of OX -modules are also OX -modules. Show that the direct sum, direct product, direct limit and inverse limit of OX -modules are OX -modules. If F and G are O H F G O L X -modules, show that omOX ( ; ) is also an X -module. Finally, if is a O F F O subsheaf of an X -module , show that the quotient sheaf L is also an X - module. O F O O F ' (5) Let (X; X ) be a ringed space and an X -module. Prove that HomOX ( X ; ) F(X). (6) Let f :(X; OX ) ! (Y; OY ) be a morphism of ringed spaces. Show that for any O F O G ∗G F ' G F X -module and Y -module we have HomOX (f ; ) HomOY ( ; f∗ ). -
SHEAVES of MODULES 01AC Contents 1. Introduction 1 2
SHEAVES OF MODULES 01AC Contents 1. Introduction 1 2. Pathology 2 3. The abelian category of sheaves of modules 2 4. Sections of sheaves of modules 4 5. Supports of modules and sections 6 6. Closed immersions and abelian sheaves 6 7. A canonical exact sequence 7 8. Modules locally generated by sections 8 9. Modules of finite type 9 10. Quasi-coherent modules 10 11. Modules of finite presentation 13 12. Coherent modules 15 13. Closed immersions of ringed spaces 18 14. Locally free sheaves 20 15. Bilinear maps 21 16. Tensor product 22 17. Flat modules 24 18. Duals 26 19. Constructible sheaves of sets 27 20. Flat morphisms of ringed spaces 29 21. Symmetric and exterior powers 29 22. Internal Hom 31 23. Koszul complexes 33 24. Invertible modules 33 25. Rank and determinant 36 26. Localizing sheaves of rings 38 27. Modules of differentials 39 28. Finite order differential operators 43 29. The de Rham complex 46 30. The naive cotangent complex 47 31. Other chapters 50 References 52 1. Introduction 01AD This is a chapter of the Stacks Project, version 77243390, compiled on Sep 28, 2021. 1 SHEAVES OF MODULES 2 In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Z- modules. Basic references are [Ser55], [DG67] and [AGV71]. We work out what happens for sheaves of modules on ringed topoi in another chap- ter (see Modules on Sites, Section 1), although there we will mostly just duplicate the discussion from this chapter. -
4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space
4. Coherent Sheaves Definition 4.1. If (X; OX ) is a locally ringed space, then we say that an OX -module F is locally free if there is an open affine cover fUig of X such that FjUi is isomorphic to a direct sum of copies of OUi . If the number of copies r is finite and constant, then F is called locally free of rank r (aka a vector bundle). If F is locally free of rank one then we way say that F is invertible (aka a line bundle). The group of all invertible sheaves under tensor product, denoted Pic(X), is called the Picard group of X. A sheaf of ideals I is any OX -submodule of OX . Definition 4.2. Let X = Spec A be an affine scheme and let M be an A-module. M~ is the sheaf which assigns to every open subset U ⊂ X, the set of functions a s: U −! Mp; p2U which can be locally represented at p as a=g, a 2 M, g 2 R, p 2= Ug ⊂ U. Lemma 4.3. Let A be a ring and let M be an A-module. Let X = Spec A. ~ (1) M is a OX -module. ~ (2) If p 2 X then Mp is isomorphic to Mp. ~ (3) If f 2 A then M(Uf ) is isomorphic to Mf . Proof. (1) is clear and the rest is proved mutatis mutandis as for the structure sheaf. Definition 4.4. An OX -module F on a scheme X is called quasi- coherent if there is an open cover fUi = Spec Aig by affines and ~ isomorphisms FjUi ' Mi, where Mi is an Ai-module. -
Essential Concepts of Projective Geomtry
Essential Concepts of Projective Geomtry Course Notes, MA 561 Purdue University August, 1973 Corrected and supplemented, August, 1978 Reprinted and revised, 2007 Department of Mathematics University of California, Riverside 2007 Table of Contents Preface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : i Prerequisites: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :iv Suggestions for using these notes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :v I. Synthetic and analytic geometry: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :1 1. Axioms for Euclidean geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. Cartesian coordinate interpretations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 3 3. Lines and planes in R and R : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 II. Affine geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1. Synthetic affine geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2. Affine subspaces of vector spaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 3. Affine bases: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :19 4. Properties of coordinate -
Math.AG] 28 Feb 2007 Osnua Leri Variety Algebraic Nonsingular a the Se[A) Let [Ma])
SOME BASIC RESULTS ON ACTIONS OF NON-AFFINE ALGEBRAIC GROUPS MICHEL BRION Abstract. We study actions of connected algebraic groups on normal algebraic varieties, and show how to reduce them to actions of affine subgroups. 0. Introduction Algebraic group actions have been extensively studied under the as- sumption that the acting group is affine or, equivalently, linear; see [KSS, MFK, PV]. In contrast, little seems to be known about actions of non-affine algebraic groups. In this paper, we show that these actions may be reduced to actions of affine subgroup schemes, in the setting of normal varieties. Our starting point is the following theorem of Nishi and Matsumura (see [Ma]). Let G be a connected algebraic group of automorphisms of a nonsingular algebraic variety X and denote by αX : X −→ A(X) the Albanese morphism, that is, the universal morphism to an abelian variety (see [Se2]). Then G acts on A(X) by translations, compatibly with its action on X, and the kernel of the induced homomorphism G → A(X) is affine. When applied to the action of G on itself via left multiplication, this shows that the Albanese morphism αG : G −→ A(G) is a surjective group homomorphism having an affine kernel. Since arXiv:math/0702518v2 [math.AG] 28 Feb 2007 this kernel is easily seen to be smooth and connected, this gives back Chevalley’s structure theorem: any connected algebraic group G is an extension of an abelian variety A(G) by a connected affine algebraic group Gaff (see [Co]) for a modern proof). The Nishi–Matsumura theorem may be reformulated as follows: for any faithful action of G on a nonsingular variety X, the induced homo- morphism G → A(X) factors through a homomorphism A(G) → A(X) having a finite kernel (see [Ma] again). -
Generalized Divisors and Reflexive Sheaves
GENERALIZED DIVISORS AND REFLEXIVE SHEAVES KARL SCHWEDE 1. Preliminary Commutative Algebra We introduce the notions of depth and of when a local ring (or module over a local ring) is \S2". These notions are found in most books on commutative algebra, see for example [Mat89, Section 16] or [Eis95, Section 18]. Another excellent book that is focussed on these ideas is [BH93]. We won't be focussing on the commutative algebra, but one should be aware, at the very least, that this background exists. In particular, we won't really use any of the theory on Cohen-Macaulay rings. However, since one ought to build up the same machinery in order to define S2, we include these definitions as well. All rings will be assumed to be noetherian. Definition 1.1. Let (R; m) be a local ring and let M be a finite R-module (which means finitely generated). An element r 2 R is said to be M-regular if rx 6= 0 for all x 2 M, x 6= 0 (in other words, r is not a zero divisor on M). A sequence of elements r1; : : : ; rn 2 R is said to be M-regular if (i) ri is a regular M=(r1; : : : ; ri−1) element for all i ≥ 1 (in particular r1 is M-regular) (ii) (r1; : : : ; rn)M ( M (that is, the containment is proper). Remark 1.2. If condition (i) is satisfied, but condition (ii) is not necessarily satisfied then such a sequence is called weakly M-regular. Remark 1.3. Notice that condition (ii) implies in a regular sequence, all ri 2 m 2 R. -
4 Sheaves of Modules, Vector Bundles, and (Quasi-)Coherent Sheaves
4 Sheaves of modules, vector bundles, and (quasi-)coherent sheaves “If you believe a ring can be understood geometrically as functions its spec- trum, then modules help you by providing more functions with which to measure and characterize its spectrum.” – Andrew Critch, from MathOver- flow.net So far we discussed general properties of sheaves, in particular, of rings. Similar as in the module theory in abstract algebra, the notion of sheaves of modules allows us to increase our understanding of a given ringed space (or a scheme), and to provide further techniques to play with functions, or function-like objects. There are particularly important notions, namely, quasi-coherent and coherent sheaves. They are analogous notions of the usual modules (respectively, finitely generated modules) over a given ring. They also generalize the notion of vector bundles. Definition 38. Let (X, ) be a ringed space. A sheaf of -modules, or simply an OX OX -module, is a sheaf on X such that OX F (i) the group (U) is an (U)-module for each open set U X; F OX ✓ (ii) the restriction map (U) (V ) is compatible with the module structure via the F !F ring homomorphism (U) (V ). OX !OX A morphism of -modules is a morphism of sheaves such that the map (U) F!G OX F ! (U) is an (U)-module homomorphism for every open U X. G OX ✓ Example 39. Let (X, ) be a ringed space, , be -modules, and let ' : OX F G OX F!G be a morphism. Then ker ', im ', coker ' are again -modules. If is an - OX F 0 ✓F OX submodule, then the quotient sheaf / is an -module. -
AG for NT Week 6
AG for NT Week 6 Recap 1. A ring, X = Spec A, M an A-module which we associated a sheaf Mf of OX -modules. Have shown: Any quasi-coherent sheafs on Spec A has this form. 2. S graded ring, X = PrjS, M graded S-module, associated to it Mf a quasi-coherent sheaf. Mf(D+(f)) = fhomogeneous elements of degree 0 in Mf g. Want to show: Given any quasi-coherent sheaf on X = ProjS, we have f =∼ Mf for some graded S-module M. We won't (can't?) say this in general, but Proposition 0.1. S graded ring, nitely generated by S1 as an S0-algebra (S = ⊕a≥0Sa). Let X = ProjS and f a quasi coherent sheaf on X. Then there is a graded S-module M such that f =∼ Mf. Remark. 1. Applies to for some ring , i.e., to n S = A[x0; : : : ; xn] A PA 2. Don't have an equivalence of categories between graded S-modules and quasi-coherent sheafs on ProjS. Indeed, 2 dierent graded modules can give the same sheaf. Example: and for some ⊕n≥0Mn ⊕n≥n0 Mn n0 ≥ 0 3. How to dene M given f? Cannot just set M = Γ(X; f) as in the ane case. Solution is to use twists. Example. , Proj r . S = A[x0; : : : ; xr] X = S = PA Claim. Γ(X; OX (n)) = Sn = fhomogenous polynomials of degree n}. In ∼ particular, Γ(X; OX ) = A. Proof. The sets D+(xi) cover X. A section t 2 Γ(X; OX (n)) is the same as an (r + 1)-tuple of sections (t0; : : : ; tr) with ti 2 OX (n)(D+(xi)) com- patible on D+(xi) \ D+(xj) = D+(xixj) Now, homogeneous elements in and OX (n)(D+(xi)) = fdeg n Sxi g OX (n)(D+(xixj)) = homogeneous elements in . -
Notes on Automorphism Groups of Projective Varieties
NOTES ON AUTOMORPHISM GROUPS OF PROJECTIVE VARIETIES MICHEL BRION Abstract. These are extended and slightly updated notes for my lectures at the School and Workshop on Varieties and Group Actions (Warsaw, September 23{29, 2018). They present old and new results on automorphism groups of normal projective varieties over an algebraically closed field. Contents 1. Introduction 1 2. Some basic constructions and results 4 2.1. The automorphism group 4 2.2. The Picard variety 7 2.3. The lifting group 10 2.4. Automorphisms of fibrations 14 2.5. Big line bundles 16 3. Proof of Theorem 1 18 4. Proof of Theorem 2 20 5. Proof of Theorem 3 23 References 28 1. Introduction Let X be a projective variety over an algebraically closed field k. It is known that the automorphism group, Aut(X), has a natural structure of smooth k-group scheme, locally of finite type (see [Gro61, Ram64, MO67]). This yields an exact sequence 0 (1.0.1) 1 −! Aut (X) −! Aut(X) −! π0 Aut(X) −! 1; where Aut0(X) is (the group of k-rational points of) a smooth connected algebraic group, and π0 Aut(X) is a discrete group. To analyze the structure of Aut(X), one may start by considering the connected automorphism 0 group Aut (X) and the group of components π0 Aut(X) separately. It turns out that there is no restriction on the former: every smooth connected algebraic group is the connected automorphism group of some normal projective variety X (see [Bri14, Thm. 1]). In characteristic 0, we may further take X to be smooth by using equivariant resolution of singularities (see e.g. -